/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 149 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 321 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 169 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 212 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 136 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 191 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (48) CpxRNTS
    (49) FinalProof [FINISHED, 0 ms]
    (50) BOUNDS(1, n^2)
(51) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 0 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 387 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 54 ms]
            (64) BOUNDS(1, INF)


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z)))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z)))

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z)))

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

The TRS has the following type information:
quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
0 :: 0:s:cons_quot
s :: 0:s:cons_quot -> 0:s:cons_quot
encArg :: 0:s:cons_quot -> 0:s:cons_quot
cons_quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
encode_quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
encode_0 :: 0:s:cons_quot
encode_s :: 0:s:cons_quot -> 0:s:cons_quot

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   quot_3
   encArg_1
   encode_quot_3
   encode_0
   encode_s_1

Due to the following rules being added:

   encArg(v0) -> 0 [0]
   encode_quot(v0, v1, v2) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]

And the following fresh constants: none

----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> 0 [0]
   encode_quot(v0, v1, v2) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
0 :: 0:s:cons_quot
s :: 0:s:cons_quot -> 0:s:cons_quot
encArg :: 0:s:cons_quot -> 0:s:cons_quot
cons_quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
encode_quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
encode_0 :: 0:s:cons_quot
encode_s :: 0:s:cons_quot -> 0:s:cons_quot

Rewrite Strategy: INNERMOST
----------------------------------------

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> 0 [0]
   encode_quot(v0, v1, v2) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   quot(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
0 :: 0:s:cons_quot
s :: 0:s:cons_quot -> 0:s:cons_quot
encArg :: 0:s:cons_quot -> 0:s:cons_quot
cons_quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
encode_quot :: 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot -> 0:s:cons_quot
encode_0 :: 0:s:cons_quot
encode_s :: 0:s:cons_quot -> 0:s:cons_quot

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = 1 + x_1
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   encode_s(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_s(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = x_1
   quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
   quot(z', z'', z1) -{ 1 }-> 0 :|: z >= 0, y >= 0, z'' = 1 + y, z1 = 1 + z, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   quot(z', z'', z1) -{ 1 }-> 1 + quot(x, 1 + z, 1 + z) :|: z'' = 0, z >= 0, z' = x, x >= 0, z1 = 1 + z


----------------------------------------

(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0


----------------------------------------

(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { quot }
   { encode_0 }
   { encArg }
   { encode_quot }
   { encode_s }

----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {quot}, {encode_0}, {encArg}, {encode_quot}, {encode_s}

----------------------------------------

(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {quot}, {encode_0}, {encArg}, {encode_quot}, {encode_s}

----------------------------------------

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {quot}, {encode_0}, {encArg}, {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: ?, size: O(n^1) [1 + z']

----------------------------------------

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 2*z'

----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {encArg}, {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']

----------------------------------------

(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {encArg}, {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']

----------------------------------------

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {encArg}, {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: ?, size: O(1) [0]

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]

----------------------------------------

(31) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]

----------------------------------------

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encArg: runtime: ?, size: O(n^1) [z']

----------------------------------------

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z'^2

----------------------------------------

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 0 }-> quot(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 0 }-> quot(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encArg: runtime: O(n^2) [2*z'^2], size: O(n^1) [z']

----------------------------------------

(37) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 2 + 2*s1 + 2*x_1^2 + 2*x_2^2 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1, s2 >= 0, s2 <= x_2, s3 >= 0, s3 <= x_3, s4 >= 0, s4 <= s1 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 2 + -4*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 2 + 2*s5 + 2*z'^2 + 2*z''^2 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= z1, s8 >= 0, s8 <= s5 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z', z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encArg: runtime: O(n^2) [2*z'^2], size: O(n^1) [z']

----------------------------------------

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 2 + 2*s1 + 2*x_1^2 + 2*x_2^2 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1, s2 >= 0, s2 <= x_2, s3 >= 0, s3 <= x_3, s4 >= 0, s4 <= s1 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 2 + -4*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 2 + 2*s5 + 2*z'^2 + 2*z''^2 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= z1, s8 >= 0, s8 <= s5 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z', z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_quot}, {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encArg: runtime: O(n^2) [2*z'^2], size: O(n^1) [z']
  encode_quot: runtime: ?, size: O(n^1) [1 + z']

----------------------------------------

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2 + 2*z' + 2*z'^2 + 2*z''^2 + 2*z1^2

----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 2 + 2*s1 + 2*x_1^2 + 2*x_2^2 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1, s2 >= 0, s2 <= x_2, s3 >= 0, s3 <= x_3, s4 >= 0, s4 <= s1 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 2 + -4*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 2 + 2*s5 + 2*z'^2 + 2*z''^2 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= z1, s8 >= 0, s8 <= s5 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z', z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encArg: runtime: O(n^2) [2*z'^2], size: O(n^1) [z']
  encode_quot: runtime: O(n^2) [2 + 2*z' + 2*z'^2 + 2*z''^2 + 2*z1^2], size: O(n^1) [1 + z']

----------------------------------------

(43) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 2 + 2*s1 + 2*x_1^2 + 2*x_2^2 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1, s2 >= 0, s2 <= x_2, s3 >= 0, s3 <= x_3, s4 >= 0, s4 <= s1 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 2 + -4*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 2 + 2*s5 + 2*z'^2 + 2*z''^2 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= z1, s8 >= 0, s8 <= s5 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z', z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encArg: runtime: O(n^2) [2*z'^2], size: O(n^1) [z']
  encode_quot: runtime: O(n^2) [2 + 2*z' + 2*z'^2 + 2*z''^2 + 2*z1^2], size: O(n^1) [1 + z']

----------------------------------------

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 2 + 2*s1 + 2*x_1^2 + 2*x_2^2 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1, s2 >= 0, s2 <= x_2, s3 >= 0, s3 <= x_3, s4 >= 0, s4 <= s1 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 2 + -4*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 2 + 2*s5 + 2*z'^2 + 2*z''^2 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= z1, s8 >= 0, s8 <= s5 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z', z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encArg: runtime: O(n^2) [2*z'^2], size: O(n^1) [z']
  encode_quot: runtime: O(n^2) [2 + 2*z' + 2*z'^2 + 2*z''^2 + 2*z1^2], size: O(n^1) [1 + z']
  encode_s: runtime: ?, size: O(n^1) [1 + z']

----------------------------------------

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z'^2

----------------------------------------

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 2 + 2*s1 + 2*x_1^2 + 2*x_2^2 + 2*x_3^2 }-> s4 :|: s1 >= 0, s1 <= x_1, s2 >= 0, s2 <= x_2, s3 >= 0, s3 <= x_3, s4 >= 0, s4 <= s1 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 2 + -4*z' + 2*z'^2 }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_quot(z', z'', z1) -{ 2 + 2*s5 + 2*z'^2 + 2*z''^2 + 2*z1^2 }-> s8 :|: s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= z1, s8 >= 0, s8 <= s5 + 1, z' >= 0, z1 >= 0, z'' >= 0
   encode_quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   encode_s(z') -{ 0 }-> 0 :|: z' >= 0
   encode_s(z') -{ 2*z'^2 }-> 1 + s9 :|: s9 >= 0, s9 <= z', z' >= 0
   quot(z', z'', z1) -{ 1 + 2*z' }-> s :|: s >= 0, s <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   quot(z', z'', z1) -{ 3 + 2*z' }-> 1 + s' :|: s' >= 0, s' <= z' + 1, z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encArg: runtime: O(n^2) [2*z'^2], size: O(n^1) [z']
  encode_quot: runtime: O(n^2) [2 + 2*z' + 2*z'^2 + 2*z''^2 + 2*z1^2], size: O(n^1) [1 + z']
  encode_s: runtime: O(n^2) [2*z'^2], size: O(n^1) [1 + z']

----------------------------------------

(49) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(50)
BOUNDS(1, n^2)

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(51) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   quot(0', s(y), s(z)) -> 0'
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))

The (relative) TRS S consists of the following rules:

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(53) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(54)
Obligation:
Innermost TRS:
Rules:
quot(0', s(y), s(z)) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
0' :: 0':s:cons_quot
s :: 0':s:cons_quot -> 0':s:cons_quot
encArg :: 0':s:cons_quot -> 0':s:cons_quot
cons_quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
encode_quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
encode_0 :: 0':s:cons_quot
encode_s :: 0':s:cons_quot -> 0':s:cons_quot
hole_0':s:cons_quot1_4 :: 0':s:cons_quot
gen_0':s:cons_quot2_4 :: Nat -> 0':s:cons_quot

----------------------------------------

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
quot, encArg

They will be analysed ascendingly in the following order:
quot < encArg

----------------------------------------

(56)
Obligation:
Innermost TRS:
Rules:
quot(0', s(y), s(z)) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
0' :: 0':s:cons_quot
s :: 0':s:cons_quot -> 0':s:cons_quot
encArg :: 0':s:cons_quot -> 0':s:cons_quot
cons_quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
encode_quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
encode_0 :: 0':s:cons_quot
encode_s :: 0':s:cons_quot -> 0':s:cons_quot
hole_0':s:cons_quot1_4 :: 0':s:cons_quot
gen_0':s:cons_quot2_4 :: Nat -> 0':s:cons_quot


Generator Equations:
gen_0':s:cons_quot2_4(0) <=> 0'
gen_0':s:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_quot2_4(x))


The following defined symbols remain to be analysed:
quot, encArg

They will be analysed ascendingly in the following order:
quot < encArg

----------------------------------------

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_0':s:cons_quot2_4(n4_4), gen_0':s:cons_quot2_4(+(1, n4_4)), gen_0':s:cons_quot2_4(1)) -> gen_0':s:cons_quot2_4(0), rt in Omega(1 + n4_4)

Induction Base:
quot(gen_0':s:cons_quot2_4(0), gen_0':s:cons_quot2_4(+(1, 0)), gen_0':s:cons_quot2_4(1)) ->_R^Omega(1)
0'

Induction Step:
quot(gen_0':s:cons_quot2_4(+(n4_4, 1)), gen_0':s:cons_quot2_4(+(1, +(n4_4, 1))), gen_0':s:cons_quot2_4(1)) ->_R^Omega(1)
quot(gen_0':s:cons_quot2_4(n4_4), gen_0':s:cons_quot2_4(+(1, n4_4)), gen_0':s:cons_quot2_4(1)) ->_IH
gen_0':s:cons_quot2_4(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(58)
Complex Obligation (BEST)

----------------------------------------

(59)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
quot(0', s(y), s(z)) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
0' :: 0':s:cons_quot
s :: 0':s:cons_quot -> 0':s:cons_quot
encArg :: 0':s:cons_quot -> 0':s:cons_quot
cons_quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
encode_quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
encode_0 :: 0':s:cons_quot
encode_s :: 0':s:cons_quot -> 0':s:cons_quot
hole_0':s:cons_quot1_4 :: 0':s:cons_quot
gen_0':s:cons_quot2_4 :: Nat -> 0':s:cons_quot


Generator Equations:
gen_0':s:cons_quot2_4(0) <=> 0'
gen_0':s:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_quot2_4(x))


The following defined symbols remain to be analysed:
quot, encArg

They will be analysed ascendingly in the following order:
quot < encArg

----------------------------------------

(60) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(61)
BOUNDS(n^1, INF)

----------------------------------------

(62)
Obligation:
Innermost TRS:
Rules:
quot(0', s(y), s(z)) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
0' :: 0':s:cons_quot
s :: 0':s:cons_quot -> 0':s:cons_quot
encArg :: 0':s:cons_quot -> 0':s:cons_quot
cons_quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
encode_quot :: 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot -> 0':s:cons_quot
encode_0 :: 0':s:cons_quot
encode_s :: 0':s:cons_quot -> 0':s:cons_quot
hole_0':s:cons_quot1_4 :: 0':s:cons_quot
gen_0':s:cons_quot2_4 :: Nat -> 0':s:cons_quot


Lemmas:
quot(gen_0':s:cons_quot2_4(n4_4), gen_0':s:cons_quot2_4(+(1, n4_4)), gen_0':s:cons_quot2_4(1)) -> gen_0':s:cons_quot2_4(0), rt in Omega(1 + n4_4)


Generator Equations:
gen_0':s:cons_quot2_4(0) <=> 0'
gen_0':s:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_quot2_4(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:cons_quot2_4(n1953_4)) -> gen_0':s:cons_quot2_4(n1953_4), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_quot2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_quot2_4(+(n1953_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_quot2_4(n1953_4))) ->_IH
s(gen_0':s:cons_quot2_4(c1954_4))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(64)
BOUNDS(1, INF)